If n is an integer which leaves remainder one when divided by three, then (1+ 3 i)^n+(1- 3 i)^n equals

Step-by-step solution

aligned & Now, (1+ 3 i)^n+(1- 3 i)^n &= [2 ( 1+ 3 i 2 ) ]^n+ [2 ( 1- 3 i 2 ) ]^n &= (-2 ^2 )^n+(-2 )^n &=(-2)^n [ ( ^2 )^ 3 r+1 +( )^ 3 r+1 ] & [ n=3 r+1, where r is an integer ] &=(-2)^n ( ^2+ )=-(-2)^n aligned